Theoretical and Computational Fluid Dynamics

, Volume 1, Issue 6, pp 303–325 | Cite as

On some control problems in fluid mechanics

  • F. Abergel
  • R. Temam
Article

Abstract

The issue of minimizing turbulence in an evolutionary Navier-Stokes flow is addressed from the point of view of optimal control. We derive theoretical results for various physical situations: distributed control, Bénard-type problems with boundary control, and flow in a channel. For each case that we consider, our results include the formulation of the problem as an optimal control problem and proof of the existence of an optimal control (which is not expected to be unique). Finally, we describe a numerical algorithm based on the gradient method for the corresponding cost function. For readers who are not interested in the mathematical details and the mathematical justifications, a nontechnical description of our results is included in Section 5.

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References

  1. C. Foias, O.P. Manley, and R. Temam (1987), Attractors for the Bénard problem: existence and physical bounds on their fractal dimension, Nonlinear Anal. TMA., 11 (8), 939–967.Google Scholar
  2. L. Landau and E. Lifschitz (1966), Mécanique des Fluides, Editions Mir, Moscow.Google Scholar
  3. J.L. Lions (1969), Contrôle Optimal des Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris. English translation, Springer-Verlag, New-York.Google Scholar
  4. D. Serre (1983), Equations de Navier-Stokes stationnaires avec données peu réguliéres, Ann. Scuola Norm. Pisa Cl. Sci. (4) 10 (4), 543–559.Google Scholar
  5. R. Temam (1984), Navier-Stokes Equations, 3rd edition, North-Holland, Amsterdam.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • F. Abergel
    • 1
  • R. Temam
    • 1
  1. 1.Laboratoire d'Analyse NumériqueCNRS et Université Paris-SudOrsayFrance

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